Maximum Likelihood Estimation for a Class of Multinomial Distributions Arising in Reliability
研究了由独立伯努利变量之和构成的多项分布模型，在可靠性实验中用于估计并联组件的失效概率，并提出了通过多项式根来识别最大似然估计的方法。
Summary Let Xi, i = 1, …,k be independent Bernoulli variables, with Xi ~ B(1, pi). Let Y = σ Xi, and consider a multinomial experiment based on n independent, identically distributed observations Y 1,…, Yn. This model is identifiable in the ordered parameter vector (p (1),…, p (k)), where p(i -1) ≤ p(i), and arises in reliability experiments in which k components in parallel have potentially different probabilities of failure. The family of multinomial distributions with the structure described above is properly contained in the family of general multinomial distributions with (k + 1) classes. Maximum likelihood estimation for this family is considered, and it is shown that for sufficiently large n, the maximum likelihood estimate of (p (1),…, p (k)) may be identified with high probability from the roots of a kth degree polynomial whose coefficients are consistent estimates of the elementary symmetric functions of the ratios θ(i) = p (i)/(1 - p (i)). A probability computation for the case k = 2 sheds light on the sample size required for our asymptotic results to take hold.